Several core concepts get confused. So before taking any deep dive into "managing risk," these need to be established.

What is Risk

There are many definitions of risk from a variety of sources. In the end these definitions are all pretty much the same.

Risk involves the probability of something happening in the future.

When that something happens it impacts the project in ways that are not good.

There can be a probability of the effect of the impact as well.

Handling the risk means knowing what kind of risk it is, and what the choices are for handling the risk.

Here's a good definition that can be used in probably any domain.

Risk refers to the uncertainty that surrounds
future events and outcomes. It is the expression
of the likelihood (probability of occurrence) and (probability of) impact of an event with the
potential to influence the achievement of an organization’s objectives. - Managing Risk in Government: An Introduction to Enterprise Risk Management, IBM Center for The Business of Government.

I've edited this a bit to remove the likelihood measure and replace it with the probability just to keep the math straight. You'll see why below when it comes to ordinal values in the risk matrix - a bad idea.

There is a time honored approach to ranking a risk, by calculating some number. This number is meanigless of course, because it has no scale, it is just a ordinalnumber. We need cardinal numbers for actual risk management.

Risk = Probability of Occurrence x Impact

This is a trick question. The risk matrix approach that results from this calculation treats these numbers as likelihood not probabilities. But in fact they are probabilities. As well they are Ordinal numbers (1, 2, 3, 4, 5 usually) and as such represents the relative ranking of the likelihood of occurrence, not the actual probability of occurrence.

This approach starts with a major problem - a show stopper problem. The two values on either side of the multiplication sign are probability distributions, not real numbers. There is no MULTIPLY operator between two probability distributions. The is the CONVOLUTION operator, but that is not likely to be of much value to ordinary project management staff. One starting point for a better understanding of the problems with the risk multiplier and risk matrix approach is the work of Tony Cox. Optimal Design of Qualitative
Risk Matrices to Classify
Quantitative Risks. Tony's source paper provides the basis for this discussion, "What's Wrong with Risk Matricies."

By category error it means the objects, the categories, do not have the proper properties to work with the operators. For some more discussion of this topic and a bit of counter discussion, see What's Right with Risk Matricies.

Where does risk come from?

Risk comes from uncertainty. There are two kinds of uncertainty.

Uncertainty that can be reduced - reducible uncertainty. This is called epistemic uncertainty. This uncertainty is characterized by the lack of knowledge. We can buy this knowledge.

Uncertainty that cannot be reduced - irreducaable uncertainty. This is called aleatory uncertainty. This uncertainty is an inherent variability in the project processes and characterized by a probability distribution. We can not buy more information about this uncertainty.

The critical point here is the reducibility and the irreducibility of these uncertainties.

If the uncertainty is irreducible, then only margin can be used to protect the project from the risk. This is scehdule margin, cost margin, techncial performance margin.

If the uncertainty is reducible, then we can buy down the uncertainty and therefore buy down the risk. That is we can spend more to find out more information - increase our knowledge. This is done in one of two ways:

Spend money in the project to increase our knowledge.

Provide a Management Reserve or Contingency to handle the risk when it comes true.

How is Risk Maanged?

We must start with a risk management process. There are several, but they are essentially the same. Here's my favorite, there are others of course, but this one covers all the steps.

So What's the Point Here?

There is no need to make up definitions for risk and risk management. There are plenty around and all of them are in use on actual projects by actual project and program managers, managing risks every day. Providing your own definitions, is sporty at best. At worst, it shows you haven't done your homework.

Uncertainty is the source of risk. There are two types of uncertainty - reducible, epistemic and irreducible, aleatory. If you do not separate these two, you cannot credibly have a risk handling plan. If you do not have a credible handling plan, then you're not actually managing risk.

Ordinal numbers cannot be used to make risk decisions. Cardinal numbers are needed. No ranking likelihood of occurrence as 1,2,3,4,5. Not allowed, it's bogus. Same for impact. Bogus. And certainly no multiplying those two, even more bogus.

Several core concepts get confused. So before taking any deep dive into "managing risk," these need to be established.

What is Risk

There are many definitions of risk from a variety of sources. In the end these definitions are all pretty much the same.

Risk involves the probability of something happening in the future.

When that something happens it impacts the project in ways that are not good.

There can be a probability of the effect of the impact as well.

Handling the risk means knowing what kind of risk it is, and what the choices are for handling the risk.

Here's a good definition that can be used in probably any domain.

Risk refers to the uncertainty that surrounds
future events and outcomes. It is the expression
of the likelihood (probability of occurrence) and (probability of) impact of an event with the
potential to influence the achievement of an organization’s objectives. - Managing Risk in Government: An Introduction to Enterprise Risk Management, IBM Center for The Business of Government.

I've edited this a bit to remove the likelihood measure and replace it with the probability just to keep the math straight. You'll see why below when it comes to ordinal values in the risk matrix - a bad idea.

There is a time honored approach to ranking a risk, by calculating some number. This number is meanigless of course, because it has no scale, it is just a ordinalnumber. We need cardinal numbers for actual risk management.

Risk = Probability of Occurrence x Impact

This is a trick question. The risk matrix approach that results from this calculation treats these numbers as likelihood not probabilities. But in fact they are probabilities. As well they are Ordinal numbers (1, 2, 3, 4, 5 usually) and as such represents the relative ranking of the likelihood of occurrence, not the actual probability of occurrence.

This approach starts with a major problem - a show stopper problem. The two values on either side of the multiplication sign are probability distributions, not real numbers. There is no MULTIPLY operator between two probability distributions. The is the CONVOLUTION operator, but that is not likely to be of much value to ordinary project management staff. One starting point for a better understanding of the problems with the risk multiplier and risk matrix approach is the work of Tony Cox. Optimal Design of Qualitative
Risk Matrices to Classify
Quantitative Risks. Tony's source paper provides the basis for this discussion, "What's Wrong with Risk Matricies."

By category error it means the objects, the categories, do not have the proper properties to work with the operators. For some more discussion of this topic and a bit of counter discussion, see What's Right with Risk Matricies.

Where does risk come from?

Risk comes from uncertainty. There are two kinds of uncertainty.

Uncertainty that can be reduced - reducible uncertainty. This is called epistemic uncertainty. This uncertainty is characterized by the lack of knowledge. We can buy this knowledge.

Uncertainty that cannot be reduced - irreducaable uncertainty. This is called aleatory uncertainty. This uncertainty is an inherent variability in the project processes and characterized by a probability distribution. We can not buy more information about this uncertainty.

The critical point here is the reducibility and the irreducibility of these uncertainties.

If the uncertainty is irreducible, then only margin can be used to protect the project from the risk. This is scehdule margin, cost margin, techncial performance margin.

If the uncertainty is reducible, then we can buy down the uncertainty and therefore buy down the risk. That is we can spend more to find out more information - increase our knowledge. This is done in one of two ways:

Spend money in the project to increase our knowledge.

Provide a Management Reserve or Contingency to handle the risk when it comes true.

How is Risk Maanged?

We must start with a risk management process. There are several, but they are essentially the same. Here's my favorite, there are others of course, but this one covers all the steps.

So What's the Point Here?

There is no need to make up definitions for risk and risk management. There are plenty around and all of them are in use on actual projects by actual project and program managers, managing risks every day. Providing your own definitions, is sporty at best. At worst, it shows you haven't done your homework.

Uncertainty is the source of risk. There are two types of uncertainty - reducible, epistemic and irreducible, aleatory. If you do not separate these two, you cannot credibly have a risk handling plan. If you do not have a credible handling plan, then you're not actually managing risk.

Ordinal numbers cannot be used to make risk decisions. Cardinal numbers are needed. No ranking likelihood of occurrence as 1,2,3,4,5. Not allowed, it's bogus. Same for impact. Bogus. And certainly no multiplying those two, even more bogus.